Question: 1 An urn contains 5 red and 2 black balls. Two balls are randomly drawn, without replacement. Let X represent the number of black balls drawn. What are the possible values of X ? Is X is random variable ? If yes, find the mean of X. Solution: … Read more
Question: In a certain collage, 4% of boys and 1% of girls are taller than 1.75 metres, Furthermore, 60% of the students in the college are girls. A student is selected at random from the college and is found to be taller than 1.75 metres. Find the probability that the … Read more
Question: Find the value of k for which the pair of equations kx + 2y = 3 and 3x + 6y = 10 has a unique solution. Solution: Given pair of linear equations is kx + 2y = 3 …….(i) 3x + 6y = 10 …….(ii) For unique solution, Hence, the pair of equations … Read more
Prove that root n is not a rational number, if n is not perfect square Solution: Let √n be a rational number. ∴ , where p and q are co-prime integers q≠ 0. On squaring both side we get ….(i) ⇒ n divides p² ⇒ n divides p … (ii) [If n divides p², then … Read more
Prove that √2 + √5 is irrational Solution: Let us suppose that √2 + √5 is rational. Let , where a and b coprime integers and b ≠ 0. Now Squaring both side ⇒ √2 is rational, since is rational. This contradict the fact that √2 is an irrational number. So, our assumption is wrong. … Read more
Prove that √p + √q is irrational, where p and q are primes Solution: Let us suppose that √p + √q is rational. Let √p + √q = a where a is rational number. ⇒ √p = a – √q On squaring both side we get ⇒ [Using formula ] The above statement is a contradiction … Read more
Prove that 2 – √ 3 is irrational, given that √3 is irrational Solution: Let 2 – √3 be a rational number. Then there exist positive integers ‘a’ and ‘ b’, where a and b are integers having no common factor other than 1 and b ≠ 0 such that … Read more
Prove that root 5 is an irrational number Solution:- Let us assume that on the contrary that √5 is a rational number. Then there exist positive integers ‘a’ and ‘ b’ such that ⇒ Where and are coprime i.e. there HCF is 1 Squaring both side ⇒ ⇒ is divisible by 5 then also divisible by … Read more
Question 2: Check whether the relation R in R defined as R={(a, b): a ≤ b³} is reflexive, symmetric or transitive. Solution:- Reflexive:- , since is not reflexive. Symmetric:- R is not symmetric. Transitive:- is not transitive. is neither reflexive nor symmetric nor transitive. Other question:- R= {(a, b): a ≤ b²} is neither reflexive … Read more
Question 1:- Show that the relation R on the set R of real numbers, defined as R= {(a, b): a ≤ b²} is neither reflexive nor symmetric nor transitive. Solution: We have, where a, b ∈ R Reflexivity:- Obiviously, is a real number and is not true. Therefore, R is not reflexive. Symmetric:- Let the … Read more